Discriminant circle bundles over local models of Strebel graphs and Boutroux curves

We study special "discriminant" circle bundles over two elementary moduli spaces of meromorphic quadratic differentials with real periods denoted by $\mathcal Q_0^{\mathbb{R}}(-7)$ and$\mathcal Q^{\mathbb{R}}_0([-3]^2)$. The space $\mathcal Q_0^{\mathbb{R}}(-7)$ is the moduli space of meromorphic quadratic differentials on the Riemann sphere with one pole of order seven with real periods; it appears naturally in the study of a neighborhood of the Witten cycle $W_5$ in the combinatorial model based on Jenkins–Strebel quadratic differentials of $\mathcal M_{g,n}$. The space $\mathcal Q^{\mathbb{R}}_0([-3]^2)$ is the moduli space of meromorphic quadratic differentials on the Riemann sphere with two poles of order at most three with real periods; it appears in the description of a neighborhood of Kontsevich's boundary $W_{1,1}$ of the combinatorial model. Applying the formalism of the Bergman tau function to the combinatorial model (with the goal of analytically computing cycles Poincaré dual to certain combinations of tautological classes) requires studying special sections of circle bundles over $\mathcal Q_0^{\mathbb{R}}(-7)$ and $\mathcal Q^{\mathbb{R}}_0([-3]^2)$. In the $\mathcal Q_0^{\mathbb{R}}(-7)$ case, a section of this circle bundle is given by the argument of the modular discriminant. We study the spaces $\mathcal Q_0^{\mathbb{R}}(-7)$ and $\mathcal Q^{\mathbb{R}}_0([-3]^2)$, also called the spaces of Boutroux curves, in detail together with the corresponding circle bundles.